Outline  A short history of spin zero ground state dominance  Present status of this Physical mechanism remains Collectivity of low-lying states by using Energy centroids of fixed spin states  Some simpler quantities can be studied for other regularities

Random matrices and random two-body interactions 1958 Wigner introduced Gaussian orthogonal ensemble of random matrices (GOE) in understanding the spacings of energy levels observed in resonances of slow neutron scattering on heavy nuclei. Ref: Ann. Math. 67, 325 (1958) 1970’s French, Wong, Bohigas, Flores introduced two-body random ensemble (TBRE) Ref: Rev. Mod. Phys. 53, 385 (1981); Phys. Rep. 299, (1998); Phys. Rep. 347, 223 (2001). Original References: J. B. French and S.S.M.Wong, Phys. Lett. B33, 449(1970); O. Bohigas and J. Flores, Phys. Lett. B34, 261 (1970). Other applications: complicated systems (e.g., quantum chaos)

Two-body random ensemble （ＴＢＲＥ） One usually chooses Gaussian distribution for two-body random interactions There are some people who use other distributions, for example, A uniform distribution between -1 and 1. For our study, it is found that these different distribution present similar statistics.

In 1998, Johnson, Bertsch, and Dean discovered that spin parity =0+ ground state dominance can be obtained by using random two-body interactions (Phys. Rev. Lett. 80, 2749) ． This result is called 0 g.s. dominance. Similar phenomenon was found in other systems, say, sd-boson systems. Ref. C. W. Johnson et al., PRL80, 2749 (1998); R.Bijker et al., PRL84, 420 (2000); L. Kaplan et al., PRB65, (2002).

An example

Some recent papers R. Bijker, A. Frank, and S. Pittel, Phys. Rev. C60, (1999); D. Mulhall, A. Volya, and V. Zelevinsky, Phys. Rev. Lett.85, 4016(2000); Nucl. Phys. A682, 229c(2001); V. Zelevinsky, D. Mulhall, and A. Volya, Yad. Fiz. 64, 579(2001); D. Kusnezov, Phys. Rev. Lett. 85, 3773(2000); ibid. 87, (2001); L. Kaplan and T. Papenbrock, Phys. Rev. Lett. 84, 4553(2000); R.Bijker and A.Frank, Phys. Rev. Lett.87, (2001); S. Drozdz and M. Wojcik, Physica A301, 291(2001); L. Kaplan, T. Papenbrock, and C. W. Johnson, Phys. Rev. C63, (2001); R. Bijker and A. Frank, Phys. Rev. C64, (R)061303(2001); R. Bijker and A. Frank, Phys. Rev. C65, (2002); L. Kaplan, T.Papenbrock, and G.F. Bertsch, Phys. Rev. B65, (2002); L. F. Santos, D. Kusnezov, and P. Jacquod, Phys. Lett. B537, 62(2002); Y.M. Zhao and A. Arima, Phys. Rev.C64, (R)041301(2001); A. Arima, N. Yoshinaga, and Y.M. Zhao, Eur.J.Phys. A13, 105(2002); N. Yoshinaga, A. Arima, and Y.M. Zhao, J. Phys. A35, 8575(2002); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev.C66, (2002); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev. C66, (2002); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, (2002); Y.M.Zhao, A. Arima, N. Yoshinaga, Phys.Rev.C66, (2002); Y. M. Zhao, S. Pittel, R. Bijker, A. Frank, and A. Arima, Phys. Rev. C66, R41301 (2002); Y. M. Zhao, A. Arima, G. J. Ginocchio, and N. Yoshinaga, Phys. Rev. C66,034320(2003); Y. M. Zhao, A. Arima, N. Yoshinga, Phys. Rev. C68, (2003); Y. M. Zhao, A. Arima, N. Shimizu, K. Ogawa, N. Yoshinaga, O. Scholten, Phys. Rev. C70, (2004); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, (2004); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C (in press) Review papers ： Y.M.Zhao, A. Arima, and N. Yoshinaga, Phys. Rep. 400, 1(2004); V. Zelevinsky and A. Volya, Phys. Rep. 391, 311 (2004).

Three interesting results  Phenomenological method by Tokyo group (namely, by us) reasonably applicable to all systems  Geometric method by GANIL group applicable to “simple” systems  Mean field method by Mexico group applicable to sd, sp boson systems

Recent Efforts  By Papenbrock & Weidenmueller by using correlation between Energy radius  By Yoshinaga & Arima & Zhao by using energy centroids and width  Hand waving ideas by a few groups (Zelevinsky, Zuker, Otsuka, and others)

Phenomenological method Let find the lowest eigenvalue; Repeat this process for all.

Probability of Imax g.s.

A few examples

Collectivity in the IBM under random interactions

Energy centroids with fixed spin

Parity distribution in the ground states  (A) Both protons and neutrons are in the shell which corresponds to nuclei with both proton number Z and neutron number N ~40;  (B) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~40 and N~50;  (C) Both protons and neutrons are in the shell which correspond to nuclei with Z and N~82;  (D) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~50 and N~82.

Conclusion and prospect  Regularities of many-body systems under random interactions, including spin zero ground state dominance, energy centroids with various quantum numbers, collectivity, etc.  Suggestion: Try any physical quantities by random interactions  Questions: parity distribution, energy centroids, constraints of collectivity, and spin 0 g.s. dominance

Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinaga (Saitama) Kengo Ogawa (Chiba) Noritake Shimizu(Tokyo) Nobuaki Yoshida (Kansai) Stuart Pittel (Delaware) R. Bijker (Mexico) J. N. Ginocchio (Los Alamos) Olaf Scholten (Groningen) V. K. B. Kota (Ahmedabad)

Empirical method by Tokyo group

d 玻色子情形

Four fermions in a single-j shell

Why P(0) staggers periodically?  对四个粒子情形，如果 GJ=-1 其他两体力为 零，Ｉ＝０的态只有一个非零的本征值．  Ｉ＝０的态的数量随 j 呈规则涨落．